This section is dedicated to the presentation of the main result of this article.
That is, the large deviation principle for stochastic differential games.
We will focus here on the case of non-cooperative games.
The cooperative case will be discussed in the last section.
The setting is exactly the same as that of [1]}.
We recall it here for the reader's convenience,
we also keep the same notation used in that article.
Throughout, we denote the set of probability measures with finite second moment on a Polish space \(E\)  by \(\mathcal {P}_2(E)\) , and we equip it with the second order Wasserstein distance denoted \(\mathcal {W}_2(\xi , \xi ^{\prime })\)  for \(\xi ,\xi ^{\prime } \in \mathcal {P}_2(E)\) , and defined as
\(\mathcal {W}_2^2(\xi , \xi ^{\prime }) := \inf _{\pi }\iint _{E\times E}d_E(x,y)\pi (dx, dy)\) 
